1.5.3 Mathematical graphs: What could they mean?
The values of the x- and y-coordinates in a graph sometimes relate to measurements of physical quantities: for example, in graphs of height against distance, or temperature against time. Physical quantities always have units associated with them, and these must be shown on the axes’ labels of the graph.
In mathematics, however, values of x- and y-coordinates that have been calculated using a formula may simply be numbers: they may not have units attached to them.
Here is an example of just such a relationship. For each value on the x-axis the corresponding value on the y-axis is given by the formula:
value of the y-coordinate = (value of the x-coordinate)^{3}_{4}
This is said as “that the value of the y-coordinate is equal to the cube of the value of the x-coordinate”. Recall that the cube of x is just the value of x multiplied by itself and by itself again.
So, for example, when the value of the x-coordinate is –3, the value of the corresponding y-coordinate is (−3)^{3}= −3 × −3 × −3 = −27. When the value of the x-coordinate is 0.4, the corresponding value of the y-coordinate is (0.4)^{3}= 0.4 × 0.4 × 0.4 = 0.064, and so on.
By convention, the value of each y-coordinate is said to depend on the value of the associated x-coordinate. That is, choose the value of an x-coordinate and then use the mathematical relationship to work out the value of the corresponding y-coordinate. The values of the x- and y-coordinates are referred to as variables, because their values are not single fixed numbers. Mathematicians sometimes call the x-coordinate the independent variable and the corresponding y-coordinate the dependent variable. But do not confuse dependence with physical causality between the associated quantities.
Table 2 Data for the cubic relationship
Value of x-coordinate | Value of y-coordinate | Coordinate pair |
---|---|---|
−3 | −27 | (−3, −27) |
−2 | −8 | |
−1 | ||
0 | ||
1 | ||
2 | ||
3 |
Activity 11: Plotting a cubic relationship
Complete Table 2 for the relationship (Y) ,which defines the y-coordinate equals the cube of the x-coordinate. Use your calculator to display the data as a line graph. What would be a suitable display window?
Without doing any calculations, describe briefly how the gradient of the graph changes as the x-coordinate changes from −3 to +3.
Discussion
Table 3 shows the completed data for the cubic relationship.
Table 3 Data for cubic relationship
Value of x-coordinate | Value of y-coordinate | Coordinate pair |
---|---|---|
−3 | −27 | (−3, −27) |
−2 | −8 | (−2, −8) |
−1 | −1 | (−1, −1) |
0 | 0 | (0, 0) |
1 | 1 | (1, 1) |
2 | 8 | (2, 8) |
3 | 27 | (3, 27) |
Figure 27 shows a sketch of the cubic relationship. Starting in the third quadrant you can see that the gradient of the curve is positive but the slope decreases as you move along the curve, closer to the point (0,0), the origin of the graph. At the origin itself, the slope of the graph is zero. As the value of the x-coordinate becomes positive, the slope begins to increase again. The curve gets steeper as you move to the right in the first quadrant. It turns out that the slope of this cubic curve is always positive, except at the origin where the slope is zero.